ScalingStacks

Definition 2.2.3.

For \(n\geq 2\) we denote by \(\operatorname{Br}_n\) the braid group with (Artin) generators \(\sigma_i\), \(1\leq i\leq n-1\). Given a generator or its inverse, we will consider the following complexes in \(\mathrm{Ch}^b(\mathrm{Sbim}_n)\): Here the part is in homological degree zero, \(m\) is induced by the multiplication map \(B_i=R \otimes_{R^{s_i}} R \langle -1 \rangle \rightarrow R\langle-1\rangle\), and \(\Delta\) is the bimodule map determined by \(1 \mapsto x_i\otimes 1 - 1\otimes x_{i+1}\). An expression \(\underline{\beta}=\sigma_{i_1}^{\epsilon_1}\cdots\sigma_{i_r}^{\epsilon_r}\) with \(\epsilon_j\in\{\pm\}\) is called a braid word with corresponding braid element \(\beta\in \operatorname{Br}_n\). The word is positive if \(\epsilon_j=1\) for \(1\leq j\leq r\). Given \(\underline{\beta}\) define \[ F(\underline{\beta}) := F(\sigma_{i_1}^{\epsilon_1}) \circ_1\cdots \circ_1F(\sigma_{i_r}^{\epsilon_r})\] where we make use of the horizontal composition \(\mathrm{Ch}^b(\mathrm{Sbim}_n)\) (given by the obvious extension of \(\circ_1=\otimes_R\)). By convention, the empty braid word gives \(F(\emptyset)=R\).